I have a Google account that I use for web history, analytics and webmaster tools. I sign into that account with my Hotmail emaill address. I am setting up a service that requires a gmail account; I do not have a gmail product on this account, so when I click on gmail, it shows: Your new address: jkotest2@gmail.com. I do not want to use jkotest2, I'm not sure how to change this. Any advice?

First and last name is there. Was your account set up initially with a non-gmail account like mine?

Nope, iirc it was always a gmail account, but depending on what I sign into that uses that email address, it mostly shows my name, but occasionally the first part of my email address instead and when it does that, it changes it on everything and I have to go into something like gtalk and change it back

See your screenshot?
Click "Products" on the left.
Next to the heading "Your Products", click "edit".
Under the heading "Delete a Product" click "Remove gmail permanently".
See if you can assign a different username.